Problem: Divide the polynomials.
Answer: Usually, there are many different ways to divide polynomials. Here, we will use the method of polynomial long division. Notice the numerator is missing a $1^{\text{st}}$ degree term. Let's add it as $0x$. $\begin{array}{r} x+2 \\ x-2|\overline{x^2+0x-5} \\ \mathllap{-(}\underline{x^2-2x\phantom{-5}\rlap )} \\ 2x-5 \\ \mathllap{-(}\underline{2x-4\rlap )} \\ -1 \end{array}$ We get that the quotient is $x+2$ and the remainder is $-1$, and therefore: $\dfrac{x^2-5}{x-2}=x+2-\dfrac{1}{x-2}$ [I want to see a different way of performing the division.]